Fractional-order chaotic systems have received increasing attention over the past few years due to their ability to effectively model memory and complexity in nonlinear dynamics. Nonetheless, most of the research conducted so far has been on constant-order formulations, which still have some limitations in terms of adaptability and reality. Thus, to evade these limitations, we present a recently designed four-dimensional hyperchaotic Chen system with variable-order fractional (VOF) derivatives in the Liouville–Caputo sense. In comparison with constant-order systems, the new system possesses excellent performance in numerous aspects. Firstly, with the use of variable-order derivatives, the system becomes more adaptive and flexible, allowing the chaotic dynamics of the system to evolve with changing fractional orders. Secondly, large-scale numerical simulations are conducted, where phase portrait orbits and time series for differences in VOF directly illustrate the effect of the order function on the system’s behavior. Thirdly, qualitative analysis is performed with the help of phase portraits, time series, and Lyapunov exponents to confirm the system’s hyperchaotic behavior and sensitivity to initial and control parameters. Finally, the model developed demonstrates a wide range of dynamic behaviors, which confirms the sufficient efficiency of VOF calculus for modeling complicated nonlinear processes. Numerous analyses indicate that this research not only shows meaningful findings but also provides thoughtful methodologies that might result in subsequent research on fractional-order chaotic systems.

This work applies two efficient techniques to investigate fractional-order systems: the Residual Power Series Method (RPSM) and the Caputo fractional derivative (CFD). Both are utilized, especially to evaluate chaotic behavior and investigate the complex dynamics associated with a four-dimensional fractional-order chaotic system. Reliable and efficient numerical simula-tions with the complexity of chaotic behavior are obtained using the CFD approach. Furthermore, derived analytical solutions of the fractional-order 4D system using the RPSM are obtained. This approach is highly preferred as it can handle several beginning conditions, is computationally efficient, and is numerically stable, hence producing rather precise results. Combining RPSM’sanalytical capacity with CFD’s strong numerical accuracy offers a comprehensive understanding of the system dynamics. Although RPSM is best for stability and simplicity of use, the CFD technique is especially helpful because of its great accuracy in predicting chaotic behavior. The suggested methods precisely define their dynamics, provide exact solutions, and clearly identify chaotic attractors. For instance, representative parameter values used in our analysis are (ν = 0.95, 0.99, 1). These results show how well CFD- and RPSM-based methods represent and solve challenging problems in engineering and scientific research.

Our goal in this paper is to use combined Laplace transform (CLT) and Adomian decomposition method(ADM) (that will be explained in section 3), to study approximate solutions for non-linear time-fractionalBurger's equation, fractional Burger's Kdv equation and the fractional modi?ed Burger's equation for theCaputo and Conformable derivatives. Comparison between the two solutions and the exact solution is made.Here we report that the Laplace transform decomposition method (LTDM) proved to be e?cient and beused to obtain new analytical solutions of nonlinear fractional di?erential equations (FDEs).

This article offers a comprehensive analysis of nonlinear time fractional Whitham-Broer-Kaup equations under Aboodh Decomposition Transform (ADT). The study model examines the effect of different fractional derivative operators on the solution behavior of these equations by comparing them with accuracy, calculation efficiency, and physical lecturer. By employing the Aboodh Transform, a mathematical tool that is powerful in solving fractional differential equations by implementing the Adomian decomposition method, we derive an approximate solutions for models assessed for specific values of the fractional order; the solutions obtained are shown in 2D and 3D. In addition, comparative analyses are performed to clarify the effect of various fractional derivative operators on the solutions achieved, which shows the accuracy and efficiency of ADT in the handling of these complex non-linear fractional partial differential equations. Furthermore, the exact and approximate solutions are compared to the constructed problem to identify the absolute errors.

This study explores the application of fractional operators to model diffusion, dispersion, dissipation, and reaction processes. Specifically, it examines the initial value problem using Caputo and conformable operators, employing the LaplaceAdomian decomposition method to derive approximate solutions. The solutions obtained are illustrated in 2D and 3D for specific values of fractional order, demonstrating their convergence. In addition, the absolute errors between the exact and approximate solutions are calculated and compared to the constructed problems. The advantages and limitations of each fractional operator are explored in terms of model accuracy, computational efficiency, and physical interpretability. This analysis reveals that conformable operator-based solutions demonstrate superior convergence rates and reduced error magnitudes compared to those obtained with the Caputo operator. In particular, the conformable solutions exhibit a more rapid convergence to the classical solution as the fractional order approaches one, suggesting a more accurate and computationally efficient representation of fractional-order dynamics. Conversely, the Caputo solutions display slower convergence and a greater deviation from the classical solution under similar conditions. These findings suggest that the conformable derivative provides a more precise and effective approximation to the classical solution, offering potential advantages for modeling fractional-order dynamics

Some times it is not easy to find the exact solution of certain partial differential equations. In this paper we use tensor product technique of Banach spaces to find certain solutions of certain fractional differential equations.